Simulation - public lecture
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The document discusses simulation as an important tool for statistical computation. It describes how simulation allows researchers to (1) reproduce randomness on computers, (2) determine probabilistic properties of statistical procedures, and (3) approximate integrals and maximize functions. The key components of simulation include pseudo-random number generators that produce uniform random variables, which are then used to simulate other probabilistic processes and systems.
![Simulation: an ubiquitous tool for statistical computation
Simulation, what’s that for?!
Illustrations
Necessity to “(re)produce chance” on a computer
Evaluation of the behaviour of a complex system (network,
computer program, queue, particle system, atmosphere,
epidemics, economic actions, &tc)
[ c Office of Oceanic and Atmospheric Research]](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-3-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Simulation, what’s that for?!
Illustrations
Necessity to “(re)produce chance” on a computer
Production of changing landscapes, characters, behaviours in
computer games and flight simulators
[ c guides.ign.com]](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-4-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Simulation, what’s that for?!
Illustrations
Necessity to “(re)produce chance” on a computer
Determine probabilistic properties of a new statistical
procedure or under an unknown distribution [bootstrap]
(left) Estimation of the cdf F from a normal sample of 100 points;
(right) variation of this estimation over 200 normal samples](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-5-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Simulation, what’s that for?!
Illustrations
Necessity to “(re)produce chance” on a computer
Approximation of a integral
[ c my daughter’s math book]](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-7-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Simulation, what’s that for?!
Illustrations
Necessity to “(re)produce chance” on a computer
Maximisation of a weakly regular function/likelihood
[ c Dan Rice Sudoku blog]](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-8-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
Pseudo-random generator
Pivotal element/building block of simulation: always requires
availability of uniform U (0, 1) random variables
[ c MMP World]](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-10-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
Pseudo-random generator
Pivotal element/building block of simulation: always requires
availability of uniform U (0, 1) random variables
Definition (Pseudo-random generator)
A pseudo-random generator is a deterministic function f from ]0, 1[
to ]0, 1[ such that, for any starting value u0 and any n, the
sequence
{u0 , f(u0 ), f(f(u0 )), . . . , f n (u0 )}
behaves (statistically) like an iid U (0, 1) sequence](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-12-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
A standard uniform generator
The congruencial generator on {1, 2, . . . , M}
f(x) = (ax + b) mod (M)
has a period equal to M for proper choices of (a, b) and becomes a
generator on ]0, 1[ when dividing by M + 1](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-20-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
A standard uniform generator
The congruencial generator on {1, 2, . . . , M}
f(x) = (ax + b) mod (M)
has a period equal to M for proper choices of (a, b) and becomes a
generator on ]0, 1[ when dividing by M + 1
Example
Take
f(x) = (69069069x + 12345) mod (232 )
and produce
... 518974515 2498053016 1113825472 1109377984 ...
i.e.
... 0.1208332 0.5816233 0.2593327 0.2582972 ...](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-21-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
A standard uniform generator
The congruencial generator on {1, 2, . . . , M}
f(x) = (ax + b) mod (M)
has a period equal to M for proper choices of (a, b) and becomes a
generator on ]0, 1[ when dividing by M + 1](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-22-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
A standard uniform generator
The congruencial generator on {1, 2, . . . , M}
f(x) = (ax + b) mod (M)
has a period equal to M for proper choices of (a, b) and becomes a
generator on ]0, 1[ when dividing by M + 1](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-23-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
The Accept-Reject Algorithm
Refinement of the above through construction of a proper box:
1. Find a hat on f , i.e. a density g that can be simulated and
such that
sup f (x) g (x) = M < ∞
x
2. Generate dots on the subgraph of g , i.e. Y1 , Y2 , . . . ∼ g , and
U1 , U2 , . . . ∼ U ([0, 1])
3. Accept only the Yk ’s such that
Uk ≤ f (Yk )/Mg (Yk )](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-33-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
The Accept-Reject Algorithm
Refinement of the above through construction of a proper box:
1. Find a hat on f , i.e. a density g that can be simulated and
such that
sup f (x) g (x) = M < ∞
x
2. Generate dots on the subgraph of g , i.e. Y1 , Y2 , . . . ∼ g , and
U1 , U2 , . . . ∼ U ([0, 1])
3. Accept only the Yk ’s such that
Uk ≤ f (Yk )/Mg (Yk )](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-34-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
The Accept-Reject Algorithm
Refinement of the above through construction of a proper box:
1. Find a hat on f , i.e. a density g that can be simulated and
such that
sup f (x) g (x) = M < ∞
x
2. Generate dots on the subgraph of g , i.e. Y1 , Y2 , . . . ∼ g , and
U1 , U2 , . . . ∼ U ([0, 1])
3. Accept only the Yk ’s such that
Uk ≤ f (Yk )/Mg (Yk )](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-35-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
Slice sampler
There exists other ways of exploitating the fundamental idea of
simulation over the subgraph:
If direct uniform simulation on
Sf = {(u, x); 0 ≤ u ≤ f (x)}
is too complex [because of unavailable hat] use instead a random
walk on Sf :](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-41-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
Slice sampler
Slice sampler algorithm
For t = 1, . . . , T
when at (x (t) , ω (t) ) simulate
1. ω (t+1) ∼ U[0,f (x (t) )]
2. x (t+1) ∼ US (t+1) , where
S (t+1) = {y ; f (y ) ≥ ω (t+1) }.](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-43-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
Slice sampler
Slice sampler algorithm
For t = 1, . . . , T
0.010
when at (x (t) , ω (t) ) simulate
0.008
1. ω (t+1) ∼ U[0,f (x (t) )]
0.006
y
2. x (t+1) ∼ US (t+1) , where
S (t+1) = {y ; f (y ) ≥ ω (t+1) }. 0.004
0.002
0.000
0.0 0.2 0.4 0.6 0.8 1.0
x](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-44-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
Slice sampler
Slice sampler algorithm
For t = 1, . . . , T
0.010
when at (x (t) , ω (t) ) simulate
0.008
1. ω (t+1) ∼ U[0,f (x (t) )]
0.006
y
2. x (t+1) ∼ US (t+1) , where
S (t+1) = {y ; f (y ) ≥ ω (t+1) }. 0.004
0.002
0.000
0.0 0.2 0.4 0.6 0.8 1.0
x](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-45-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
Slice sampler
Slice sampler algorithm
For t = 1, . . . , T
0.010
when at (x (t) , ω (t) ) simulate
0.008
1. ω (t+1) ∼ U[0,f (x (t) )]
0.006
2. x (t+1) ∼ US (t+1) , where
y
0.004
S (t+1) = {y ; f (y ) ≥ ω (t+1) }. 0.002
0.000
0.0 0.2 0.4 0.6 0.8 1.0
x](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-46-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
Slice sampler
Slice sampler algorithm
For t = 1, . . . , T
0.010
when at (x (t) , ω (t) ) simulate
0.008
1. ω (t+1) ∼ U[0,f (x (t) )]
0.006
2. x (t+1) ∼ US (t+1) , where
y
0.004
S (t+1) = {y ; f (y ) ≥ ω (t+1) }. 0.002
0.000
0.0 0.2 0.4 0.6 0.8 1.0
x](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-47-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
Slice sampler
Slice sampler algorithm
For t = 1, . . . , T
0.010
when at (x (t) , ω (t) ) simulate
0.008
1. ω (t+1) ∼ U[0,f (x (t) )]
0.006
2. x (t+1) ∼ US (t+1) , where
y
0.004
S (t+1) = {y ; f (y ) ≥ ω (t+1) }. 0.002
0.000
0.0 0.2 0.4 0.6 0.8 1.0
x](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-48-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
Slice sampler
Slice sampler algorithm
For t = 1, . . . , T
0.010
when at (x (t) , ω (t) ) simulate
0.008
1. ω (t+1) ∼ U[0,f (x (t) )]
0.006
2. x (t+1) ∼ US (t+1) , where
y
0.004
S (t+1) = {y ; f (y ) ≥ ω (t+1) }. 0.002
0.000
0.0 0.2 0.4 0.6 0.8 1.0
x](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-49-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
Slice sampler
Slice sampler algorithm
For t = 1, . . . , T
0.010
when at (x (t) , ω (t) ) simulate
0.008
1. ω (t+1) ∼ U[0,f (x (t) )]
0.006
2. x (t+1) ∼ US (t+1) , where
y
0.004
S (t+1) = {y ; f (y ) ≥ ω (t+1) }. 0.002
0.000
0.0 0.2 0.4 0.6 0.8 1.0
x](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-50-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
The Metropolis–Hastings algorithm
Further generalisation of the fundamental idea to situations when
the slice sampler cannot be easily implemented
‘local’ exploration can produce ‘global’ vision:
[ c Civilization 5]](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-53-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
Properties
Always accepts higher point and sometimes lower points
(similar to gradient algorithms)
Depends on the dispersion de g
Average probability of acceptance
= min{f (x), f (y )}g (y − x) dxdy
close to 1 if g has a small variance [Danger!]
far from 1 if g has a large variance [Re-Danger!]](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-60-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
Properties
Always accepts higher point and sometimes lower points
(similar to gradient algorithms)
Depends on the dispersion de g
Average probability of acceptance
= min{f (x), f (y )}g (y − x) dxdy
close to 1 if g has a small variance [Danger!]
far from 1 if g has a large variance [Re-Danger!]](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-61-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Producing randomness by deterministic means
Properties
Always accepts higher point and sometimes lower points
(similar to gradient algorithms)
Depends on the dispersion de g
Average probability of acceptance
= min{f (x), f (y )}g (y − x) dxdy
close to 1 if g has a small variance [Danger!]
far from 1 if g has a large variance [Re-Danger!]](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-62-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Monte Carlo principles
General purpose of Monte Carlo methods
Given a probability density f known up to a normalizing constant,
˜
f (x) ∝ f (x), and an integrable function h, compute
˜
h(x)f (x)dx
I(h) = h(x)f (x)dx =
˜
f (x)dx
when ˜
h(x)f (x)dx is intractable.
[Remember π!!]](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-66-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Monte Carlo principles
General purpose of Monte Carlo methods
Given a probability density f known up to a normalizing constant,
˜
f (x) ∝ f (x), and an integrable function h, compute
˜
h(x)f (x)dx
I(h) = h(x)f (x)dx =
˜
f (x)dx
when ˜
h(x)f (x)dx is intractable.
[Remember π!!]](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-67-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Simulated annealing
Optimisation problems
A (genuine) puzzle
During a dinner with 20 couples sitting at four tables with ten
seats, everyone wants to share a table with everyone. The
assembly decides to switch seats after each serving towards this
goal. What is the minimal number of servings needed to ensure
that every couple shared a table with every other couple at some
point? And what is the optimal switching strategy?
[http://xianblog.wordpress.com/2012/04/12/not-le-monde-puzzle/]
solution to the puzzle](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-74-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Simulated annealing
Resolution via simulation
The simulated annealing algorithm:
name is borrowed from metallurgy
metal manufactured by a slow
decrease of temperature
(annealing) stronger than when
manufactured by fast decrease
[ c Joachim Robert, ca. 2006]](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-89-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Simulated annealing
Resolution via simulation
Repeat
Random modifications of parts of the original circuit with cost
C0
Evaluation of the cost C of the new circuit
Acceptation of the new circuit with probability
C0 − C
exp ∧1
T
[Metropolis et al., 1953]](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-90-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Simulated annealing
Resolution via simulation
Repeat
Random modifications of parts of the original circuit with cost
C0
Evaluation of the cost C of the new circuit
Acceptation of the new circuit with probability
C0 − C
exp ∧1
T
T , temperature, is progressively reduced
[Metropolis et al., 1953]](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-91-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Simulated annealing
A family meeting (1)
Recall the table puzzle
Definition of a target function
T=sample(rep(1:4,5))
for (i in 2:6)
T=cbind(T,sample(rep(1:4,5)))
meet=function(T){
M=outer(T[,1],T[,1],"==")
for (i in 2:6)
M=M+outer(T[,i],T[,i],"==")
M
}
penalty=function(M){ sum(M==0) }
penat=penalty(meet(T))](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-93-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Simulated annealing
A family meeting (2)
Random switch of couples
Pick two couples [among the 20 couples] at random with
probabilities proportional to the number of other couples they
have not seen
prob=apply(meet(T)==0,1,sum)
switch their respective position during one of the 6 courses
accept the switch with Metropolis–Hsatings probability
log(runif(1))<(penalty(old)-penalty(new))/gamma](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-94-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Simulated annealing
A family meeting (2)
Random switch of couples
for (t in 1:N){
prop=sample(1:20,2,prob=apply(meet(T)==0,1,sum))
cour=sample(1:6,1)
Tp=T
Tp[prop[1],cour]=T[prop[2],cour]
Tp[prop[2],cour]=T[prop[1],cour]
penatp=penalty(meet(Tp))
if (log(runif(1))<(penat-penatp)/gamma){
T=Tp
penat=penatp}
}](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-96-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Simulated annealing
A family meeting (3)
Solution
T
[1,] 1 4 3 2 2 3
[2,] 1 2 4 3 4 4
[3,] 3 2 1 4 1 3
[4,] 1 2 3 1 1 1
[5,] 4 2 4 2 3 3
[6,] 2 4 1 2 4 1
[7,] 4 3 1 1 2 4
[8,] 1 3 2 4 3 1
[9,] 3 3 3 3 4 3
[10,] 4 4 2 3 1 1
[11,] 1 1 1 3 3 2
[12,] 3 4 4 1 3 2
[13,] 4 1 3 4 4 2
[14,] 2 4 3 4 3 4
[15,] 2 3 4 2 1 2
[16,] 2 2 2 3 2 2
[17,] 2 1 2 1 4 3
[18,] 4 3 1 1 2 4
[19,] 3 1 4 4 2 1
[20,] 3 1 2 2 1 4 [ c http://www.metrolyrics.com]](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-97-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Simulated annealing
Solving sudokus
Solving a Sudoku grid as a minimisation problem:
[ c Dan Rice Sudoku blog]](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-98-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Simulated annealing
Solving sudokus
Given a partly filled Sudoku grid (with a single solution),
define a random Sudoku grid by filling the empty slots at
random
s=matrix(0,ncol=9,nrow=9)
s[1,c(1,6,7)]=c(8,1,2)
s[2,c(2:3)]=c(7,5)
s[3,c(5,8,9)]=c(5,6,4)
s[4,c(3,9)]=c(7,6)
s[5,c(1,4)]=c(9,7)
s[6,c(1,2,6,8,9)]=c(5,2,9,4,7)
s[7,c(1:3)]=c(2,3,1)
s[8,c(3,5,7,9)]=c(6,2,1,9)](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-99-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Simulated annealing
Solving sudokus
Given a partly filled Sudoku grid (with a single solution),
define a random Sudoku grid by filling the empty slots at
random
define a penalty function corresponding to the number of
missed constraints
#local score
scor=function(i,s){
a=((i-1)%%9)+1
b=trunc((i-1)/9)
boxa=3*trunc((a-1)/3)+1
boxb=3*trunc(b/3)+1
return(sum(s[i]==s[9*b+(1:9)])+
sum(s[i]==s[a+9*(0:8)])+
sum(s[i]==s[boxa:(boxa+2),boxb:(boxb+2)])-3)
}](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-100-320.jpg)
![Simulation: an ubiquitous tool for statistical computation
Simulated annealing
Solving sudokus
Given a partly filled Sudoku grid (with a single solution),
define a random Sudoku grid by filling the empty slots at
random
define a penalty function corresponding to the number of
missed constraints
fill “deterministic” slots
make simulated annealing moves
# random moves on the sites
i=sample((1:81)[as.vector(s)==0],sample(1:sum(s==0),1,pro=1/(1:sum(s==0))))
for (r in 1:length(i))
prop[i[r]]=sample((1:9)[pool[i[r]+81*(0:8)]],1)
if (log(runif(1))/lcur<tarcur-target(prop)){
nchange=nchange+(tarcur>target(prop))
cur=prop
points(t,tarcur,col="forestgreen",cex=.3,pch=19)
tarcur=target(cur)
}](https://image.slidesharecdn.com/simulation-publiclecture-120710074937-phpapp01/85/Simulation-public-lecture-101-320.jpg)
Simulation - public lecture
- 1. Simulation: an ubiquitous tool for statistical computation Simulation: an ubiquitous tool for statistical computation Christian P. Robert Universit´ Paris-Dauphine, IUF, & CREST e http://www.ceremade.dauphine.fr/~xian July 9, 2012
- 2. Simulation: an ubiquitous tool for statistical computation Outline Simulation, what’s that for?! Producing randomness by deterministic means Monte Carlo principles Simulated annealing
- 3. Simulation: an ubiquitous tool for statistical computation Simulation, what’s that for?! Illustrations Necessity to “(re)produce chance” on a computer Evaluation of the behaviour of a complex system (network, computer program, queue, particle system, atmosphere, epidemics, economic actions, &tc) [ c Office of Oceanic and Atmospheric Research]
- 4. Simulation: an ubiquitous tool for statistical computation Simulation, what’s that for?! Illustrations Necessity to “(re)produce chance” on a computer Production of changing landscapes, characters, behaviours in computer games and flight simulators [ c guides.ign.com]
- 5. Simulation: an ubiquitous tool for statistical computation Simulation, what’s that for?! Illustrations Necessity to “(re)produce chance” on a computer Determine probabilistic properties of a new statistical procedure or under an unknown distribution [bootstrap] (left) Estimation of the cdf F from a normal sample of 100 points; (right) variation of this estimation over 200 normal samples
- 6. Simulation: an ubiquitous tool for statistical computation Simulation, what’s that for?! Illustrations Necessity to “(re)produce chance” on a computer Validation of a probabilistic model Histogram of 103 variates from a distribution and fit by this distribution density
- 7. Simulation: an ubiquitous tool for statistical computation Simulation, what’s that for?! Illustrations Necessity to “(re)produce chance” on a computer Approximation of a integral [ c my daughter’s math book]
- 8. Simulation: an ubiquitous tool for statistical computation Simulation, what’s that for?! Illustrations Necessity to “(re)produce chance” on a computer Maximisation of a weakly regular function/likelihood [ c Dan Rice Sudoku blog]
- 9. Simulation: an ubiquitous tool for statistical computation Simulation, what’s that for?! Illustrations Necessity to “(re)produce chance” on a computer Pricing of a complex financial product (exotic options) Simulation of a Garch(1,1) process and of its volatility (103 time units)
- 10. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Pseudo-random generator Pivotal element/building block of simulation: always requires availability of uniform U (0, 1) random variables [ c MMP World]
- 11. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Pseudo-random generator Pivotal element/building block of simulation: always requires availability of uniform U (0, 1) random variables 0.1333139 0.3026299 0.4342966 0.2395357 0.3223723 0.8531162 0.3921457 0.7625259 0.1701947 0.2816627 . . .
- 12. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Pseudo-random generator Pivotal element/building block of simulation: always requires availability of uniform U (0, 1) random variables Definition (Pseudo-random generator) A pseudo-random generator is a deterministic function f from ]0, 1[ to ]0, 1[ such that, for any starting value u0 and any n, the sequence {u0 , f(u0 ), f(f(u0 )), . . . , f n (u0 )} behaves (statistically) like an iid U (0, 1) sequence
- 13. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Pseudo-random generator Pivotal element/building block of simulation: always requires availability of uniform U (0, 1) random variables q 0.8 q q 0.6 xt+1 q 0.4 q q 0.2 q q q 0.0 0.0 0.2 0.4 0.6 0.8 10 steps (ut , ut+1 ) of a uniform generator
- 14. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Philosophical foray ¡Paradox! While avoiding randomness, the deterministic sequence (u0 , u1 = f(u0 ), . . . , un = f(un−1 )) must resemble a random sequence! Debate on whether or not true randomness does exist (Laplace’s demon versus Schroedinger’s cat), in which case pseudo random generators are not random (von Neuman’s state of sin)
- 15. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Philosophical foray ¡Paradox! While avoiding randomness, the deterministic sequence (u0 , u1 = f(u0 ), . . . , un = f(un−1 )) must resemble a random sequence! Debate on whether or not true randomness does exist (Laplace’s demon versus Schroedinger’s cat), in which case pseudo random generators are not random (von Neuman’s state of sin)
- 16. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Philosophical foray ¡Paradox! While avoiding randomness, the deterministic sequence (u0 , u1 = f(u0 ), . . . , un = f(un−1 )) must resemble a random sequence! Debate on whether or not true randomness does exist (Laplace’s demon versus Schroedinger’s cat), in which case pseudo random generators are not random (von Neuman’s state of sin)
- 17. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Philosophical foray ¡Paradox! While avoiding randomness, the deterministic sequence (u0 , u1 = f(u0 ), . . . , un = f(un−1 )) must resemble a random sequence! Debate on whether or not true randomness does exist (Laplace’s demon versus Schroedinger’s cat), in which case pseudo random generators are not random (von Neuman’s state of sin)
- 18. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means True random generators Intel circuit producing “truly random” numbers: There is no reason physical generators should be “more” random than congruential (deterministic) pseudo-random generators, as those are valid generators, i.e. their distribution is exactly known (e.g., uniform) and, in the case of parallel generations, completely independent
- 19. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means True random generators Intel generator satisfies all benchmarks of “randomness” maintained by NIST: Skepticism about physical devices, when compared with mathematical functions, because of (a) non-reproducibility and (b) instability of the device, which means that proven uniformity at time t does not induce uniformity at time t + 1
- 20. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means A standard uniform generator The congruencial generator on {1, 2, . . . , M} f(x) = (ax + b) mod (M) has a period equal to M for proper choices of (a, b) and becomes a generator on ]0, 1[ when dividing by M + 1
- 21. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means A standard uniform generator The congruencial generator on {1, 2, . . . , M} f(x) = (ax + b) mod (M) has a period equal to M for proper choices of (a, b) and becomes a generator on ]0, 1[ when dividing by M + 1 Example Take f(x) = (69069069x + 12345) mod (232 ) and produce ... 518974515 2498053016 1113825472 1109377984 ... i.e. ... 0.1208332 0.5816233 0.2593327 0.2582972 ...
- 22. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means A standard uniform generator The congruencial generator on {1, 2, . . . , M} f(x) = (ax + b) mod (M) has a period equal to M for proper choices of (a, b) and becomes a generator on ]0, 1[ when dividing by M + 1
- 23. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means A standard uniform generator The congruencial generator on {1, 2, . . . , M} f(x) = (ax + b) mod (M) has a period equal to M for proper choices of (a, b) and becomes a generator on ]0, 1[ when dividing by M + 1
- 24. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Approximating π My daughter’s pseudo-code: N=1000 π=0 ˆ for I=1,N do X=RDN(1), Y=RDN(1) if X2 + Y2 < 1 then π = π+1 ˆ ˆ end if end for return 4*ˆ /N π
- 25. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Approximating π pi = 3.2 q q q q My daughter’s pseudo-code: q q q q q q q q q q q q q N=1000 q q q q q q q q q qq q q q π=0 ˆ q q q q q q q qq q q q q q q for I=1,N do q q q q q q q q X=RDN(1), Y=RDN(1) q q q q q q q q q q if X2 + Y2 < 1 then q q q q q q qq q q q q q q q π = π+1 ˆ ˆ q q q q q q q q q qq end if q q q q q q q q q q end for return 4*ˆ /N π 100 simulations
- 26. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Approximating π pi = 3.108 q q q q q q q q qq q q qq q q qq q q q q q q q q q q q q q qq q qq q q q q q q qq q q q qq q q My daughter’s pseudo-code: q q q q q q qq q q q q q q q qq qq q q q q q q q q q q q q q q qq q q q q q q qq q q qq q q q q q q q q q q qq q q qq q q qqq q q q q q q q q q qq q q q q q q q q q qq q q q qq q q q q qq q q qq q q q q q q qq qq q q q q q q q N=1000 q q q q qq q q q qq q q qq q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q qq q qq q q qq q q q q q q qq qq q q q q q qq qq q q q q q q qq q q q q q q q q q q q q q q q q qq q qq qq qq q q q q q q qq q π=0 ˆ q q q q q q q q q qq qq q q qq q q q q q q q q qq q qq q q q q q q qq q q q q q q q q qqq q qq qq q q q q q q q q q q q q qq q q q q q q q q q qq q q q qq qq q q q q q q q q q q qq q q q q qq q q q q q qqq q q for I=1,N do q qq q qq q q q qq q q qqqq q q q q q q q q q q q q q q qq q q q q q q q qq q q q qq q q q qq qq q q q q qq q q q qq q qq q q q q q qq q qq q q q q q q q qq q q q q qq qq q q q qq q q qq qq q q q qq q q q q q qq q q q q q q q q q X=RDN(1), Y=RDN(1) q q q q q q q q q q q q qq q q q q q q q q qq q q q qq qq q q q q qq qq q q q q q q qq q qq q q q q qq qq q q q q q q qq q qq q q q qq q q qq q q q q q if X2 + Y2 < 1 then q q q qq q q qq q q q q q q q qq q q qq q q q q q qq q qqq q q q qq q q q q q qq qq q q q q q q q q q q q q q q q q qq q q q qq q qq q q q q q q q q q q q q qq q q q qq qq q q q q q qq q q qqq q q q q q q q q q q q qq q q q q q q q qq q q q qq q q q q q q π = π+1 ˆ ˆ q qq q q q q q q q q qq q q qq q q q q q q q qq q q q q q q q q q q qq q q qq q q q qq q q qq q qq q q q qq q q q q q q qq q qq q qq q q q q q q q q qq q qq q q q q q qq qq q q q q q qq q q q qq q q q q qq q q q q q q q q q q qq qq q qq end if q qq q q q q q q q q q q q q q q qq q q q q q qq q qq q q q q q qq q q qq q qq q q q qq q q q q qq q q q qq q q q q q q q q q q q qq q q q q q q q q qq q q q q qqq q q q qq q q end for return 4*ˆ /N π 1000 simulations
- 27. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Approximating π pi = 3.136 qqq q q q q q q q qq qq qq qq q q qq q qq q q q qqqq qq qq qq q q qqq q qq qqqq q qq q q qq qqq qq qq qqq q q q q q qq q q qqq qq q q qq qq q qq q q q q q q qq q q q q qq q qq q qq q q q q qq q q qq q q qq q qq qq q q qq q q q q qqq qq qq q q q q qq q q q qq qq q q qqq q q q q q q qq q qq q q q q q qq q q qq q qq q q q q qq q q q q q q qq q qq q qq q q q qq qq qq q qqqq qqqq q qqq qq q q qq q q qqq q q qq qqq q qqqqqq q qqqq q qqqq qq q qqqq qqqqq qq qq qqq qqq qqq qqqqq q qqqqqqq q q qqqq q q q qq q q q q q qqqq q qq q q qq q qqq qq qq qq q q q q q qq q qqqq q q q q q q q q q q qq qq q qq qq qqqq q q qqqqqq q qqq q qqq qqqq qq q qqq qqqqqq qqq qq q q qq qq qqq qqqq qq q q q q q qq q q q qqq qq q q q qqqqq q q qq q q qq q q q qq q q qq qq q qq q q qq q q q q q q q q qq qqqqq qqq q qqqqq qq q q qq q qqq qq q qq q q qq qq q q q q qq q q q q q qq q qq qq qqqqq qq qq qqqqqqqq qqqqq qqq qqqq qq q q qqqqqq qqqq q qqqqqqq qqqqq q q q q qqq q q q q qq q q q q qq q q q q q qq qq qqq qqqq q qqqq qq q q q q qq q q qq q q q q q q q q q q qq q q q q q q q q q q q qq q q q qq q qq q My daughter’s pseudo-code: qq qq q qq q q q qq qq q qqq q qq qq q q qq qq q qq q q q q q q q q q qq q qq qq qq q q qq q q q q qq q qq q q q q q qqq q q qq qq q qq qq q qq qq qq q q q q qq qqq qq qqq q q qq qqqq q qqqqqqq q q q q q q q q q q q q qq q q q q q q qqq q q q q q q q qq qq q q q q q q qq q q q q qq qqqqq qqqqqq q qqqqqq qq q q q qqq qq q q qqqq q qq qqq q q qq q q q q q qqq qqqq q qq qq q q qqq q q q q qqq qq q q qq qqqq q q q q q qqqq qqqq q q qq qq q q q qqqqqq qqqq qqqq q q q q q q q qq q q qq qq q qq q q q q qq qqq q qq qq qq q qq q qqq q qqqqq qq q q q q q q qqq q qq qq qq qq q q q qq qq q q q q q q q q qqq qqq q qq q q qq qq qq q q qqq q q q q q q qq q q qq q qq q qqq q q q q q q qq q q q q q q q qq q q q q q q q qq q q q qq q q q q q qq q q q qqq q qq q q q q q qqq q q qq q q q qq q q qq q q qq q q q q qq q q q qq qqqqqqq qq q q qq q q q qq qq q q q q q q qqqqq q qqqqq q qq qqq q q qqq q qq qqqqq qqqqqqqqqqqq q q qq qq q qqq qqqqqqqqq qq q q q qq q q q qq q q q q q q q q q q qq q qq q q q q qq q q q q q qqqqq q qq qq qqq qqqq qq qqqqqq q q qq q qqqq qqq qqqq q q q qqqq qq q q q q q qqq q q q q qq q q q q q qqq q q q q q q q q qqq q q qq q q qqq q qq q q q q q q qq qq q q qqq q q q q q q q qq q q qq q qq q q q qq qq qqq qq qq qq qqq q q q q q qqq q qq q q q q q q qq q q q q qqq q q q q q q qq q q qq qqqq qq q qq q q qq q qq q qq q q q q q q q q q qq qq q q q qqq q q qq q q q q q qq q q qqq qqq qqqqqq qqq q q q q q q q q qq qqq q q qq q q q qq q q qq q q q q qq q q q q q qq q q q q q q qq q q q qq qqq qq qqq qq qqqqq qq q q q q q qq q qq q q q qq qqqq qqq q q qqq qq q qqq q q q q q q qqqqq q qqq q qq q qq qqqq q qqqqq qq qq q qqqqqq qqqqqq q qqqqqqq q q qq q qqqq qq q q q qq qqqq qqq qq q q q q q q q q q qq q qq q qqq qq q q q q q q q qqq q qq qqq qqq q qq qqqqq qq q q q qq qq q q q q qq qqq q q q q q q N=1000 qq q q q qq qq q q qq qqq qq q q qq qq qqqq q q qq q q q q qq q q q q q q q q qq q qq q q q qqq q qq q qq qq q q q q qqq q qq qq q qq qq q q qq q qq qq q q qq q qqqq q q qq q qq q qq q q q q q q qq q q q qq q q q q q qq q q q q qqq q qq qq qqq qq q q q qq qq q q q q q q q q qq qq qqqqqqq q q qq qqqq qqqq q qqqqqqq qq qqq qqqqqqqq qqq qqq q qqq qqqqqq q q q qq qq q q qq qq qqqqq qq qqqq qq q q qq q qqqq q q q q qq q q qq q q qqqq q qq q qqq q qqq qq qq qqq q q q q q q q q q qq q q q q q q q q q qqq qqq q q qqqqq qqq q qqq q q qq qqqq qqqq qqqqq q qqqqq qq qqqqqqq q qqq q q q q q qq qqq q q q q q q qqq q q qqq qq q q qq q q qqqq q q qqq q q q q q qq q q q qq qq qq q q q qq q qq q qq q qqq q q q q q q qq qq q q q q q q q q q qqq q q q q q qq q qq q qq qq qqq q q qq q q q qq q qqqq qq q q q qqq q q q q qqq q qqq q qq q q qq q qq q qq qqq qqqqq q q qqqqqqqqq q q q qq qq q qq q q qqqqq qqqqqq q qqqqqqqqqq q qqqqqq qq q q 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qqq qqqqqqqqq q qqqqq qqqqqqqq q q qq qqqqq q q q q q q q qqq qq q qq qqq q qq qqqqq qqqqqqq q qqq q q qqq qqq qq q qq q q qqq q q q q q q qqq q q q q qq q q qq q q q q qq q q q q q qq q q q for I=1,N do qqqqq qq qq q qqq q q qq q qqqqq q qqqq q q qq q qqq q q q qqqqq qqqqq q qqqqq q qq qq q q qqq q q q q q qqqqq q q q q qq q q qq qq q qq qq q qq qq q q qqqq qq q qq q qq qqq qq q q q q q q qqq q qqq qq q qqqq q q qqq qqq q q q qq qq q q q q qq q qq q q qqqq q q q qqq q q q q qq qqq qq qq q q q q qq qqq q qq q qq q q qq q q qq q q q q q qq q qq q qqqq q q q q q q q q q q q q qq qqq q q q q qq q q q q qq q q q qq qq q q qqqqqq q qq q qq qqq q q q qq qq q qq q q qqqqqq qq q qq qqq qqq qqq q q qq q q qq q qq q q q q q qq q q q q qq q qq q q q qq qq q q qq q q q qq q q q q qq q q q q qq q q qq q qq q q q q q qqq qq q q q q q qq q qqq qq q qqq q q q qqq qqq qqq q q qqqqqqq qqqqq q q qqq qqq q q qq q q q q q q q q q qq q qqq q q q qq qq qq q q qq q q q q qq q qqq qq q q qqqqqq q qq qqq q q qq q qq qq q q qqqqqqqqqqq q qqqqqqqqqqqq q q q q q q q q q qqqq qq qq q q qqqq qq q q q q qq qq qq q q qq q qq qq qq q qqq qq qqqqqq q qq qqqq qqqq q qq q q q q q q q qq qqq q qqq q qq q q qqqq q qqqq qqqqqq qqqq q q q q q q qq q q q q q q q qq qqq q q q q qqq q qqq q q qqq q qq qq qq q q q q qq qq q qq q q q q q q qq q q q qq qq qq qq q q q q q qq q q qq q q qq q q q q qq qq qq q qq q q q q q q qq q q qq qq qq q q qq q q qq q q qqq q q q qqq q q qqqq q q q qqqqq qqqq qq qqqqq q q qqqqqqqq qqqq q q qq q q qqqqqqq qqq q qq q qqq q qq qqq qqqqqq qqq q q q q qqqq qqq qqqqq qq qq q qq qq qq q q q q q q q qq q q q q q q q q qq q qqq q q q qq q qq q qq q qqqq qqqqqq qq qq qq qqqq qqq q qq qqq qqqqq qqq q q qq qqq qqqq qqqqqqq q q q q q qq q q q q q q q q q qq q q qq q qq q qq q q qq q qq q q q q qq q q q q qq q q qq q q q q q q q q q q qqq q q q qq q q q qqq q qqq q q q q qqq qq qq qq qqqqq qq q q q q X=RDN(1), Y=RDN(1) q q q qq qqq qqqqq qq qqqq qq qqqqq qqqqqq qq qqq qqq qqq qq qq qqq qqq qq q qqqq q qqqq qq q q q qq qq q qq q q qq qqq qqq q q qqq q qq qqqqqq qqq qq q q q q q qqq q qq q q q qq q q q q qq q q q q q q q q qq q q qq q qqqqqq q qqq qqqq q q q q qq qqq qq q q q q qq q qq q qq qq q q q q q qq q q q q q qq q qq q q q qqq q qq q q q q q q qq q q q qq q q qq qq qq qq q qq qq q q q q q q qqq qq q q q q q qq q qq q q qq qq qqq q q q qqq qqq qqqq qq qq q q q qqqq q qq q q q q q q q q q q q q qq q q q qq qqq qqq q q q q qqqq qq q q qqqq q q qqq q qqqq q q qqqqqqq q qq q q qqq q qqq qq q q q q qq qq qqq q q q qqqqqqq qq q qq qqqq q qq q q q qq q qqq qqq q q qq q q q q q qqq q q q q qq q q q q q qqqqq qq q q q qq q qqq q q q qq q q q q q q q q qq qq q qqq q qq qq q q qq qqq q qq qqqqq q q q qq q qq q qqq q qqqq q qqq q qqq q qq qqqq qq q q qq q q qqq q qq q q q q qqq q qq qq q q qqqq qq qq qqqqqq qq q qq q qqqq qq q qqqqq q qq qq q qq qqqq qq qqqq q q q qqqq qq qqq q q q qq qq q q q q qq q qq q q q qq q q q q q qq qq q qq qq qq q qq q q q q qqq q q q q qq q q q q qq qq q q q q q qq qq q q q q q q q q qq q q q qq q q q q q q q q q qqq qq q q qq q q q q q q q q qq q qq q qqqqqq q qqqq q qqqqqq qq q qqqq qqqqqq q qqqqqqq q qq qqq q qqqq qqq qqqqqq q qq if X2 + Y2 < 1 then q q q q q qqqq q q q qq qqq qqqq q q qqq qqqq q qqq qqq q q qqqq q q qq qq q q q qq q qq q qq q q q q qq qqq q q q q qqq q q q qq qqqq qqqq q q qq q q q qq q qq q q q qq q q q qq qq qq qq q q q q q q qq q q q q q qq q q q q q qq q q q qqqqq qq qq q q q qqq qqqqqq qqqqqqqq q qqqqq q q q qqq qqqq q qqq qqq q q qqqq qqq qqq q qq q qq q q q qqq q qq qqq qqqq q q qqqq q qqqqq q q qqqqqqqqqqq q qq qqq qqqq qqq qqq q q qqqqqq qq q qq qqq q q q q qqq q qq q q q qq q qq q qqqqqqq q q qq q qqq q qq qqq qqqqq qq q q qq q qq qq qqqqq qqqq qq q q qq q q q qq qq q qq qqqqqq q qqqqqq qq qq q qq q q q q q q q qq q q q q q q q q q q q qq qq q qq q q qqqqq qqq q q q q qq q q q qqq q q qqqqqq q q q qq qq q q qqqq q q q q q qqqq q q q q q q q q qq q qqqqq q q q qq q qq q q q q q q q q q qqq q q q q qq q q q q q q qq q q qq q q qq q qqq qq qq q q q q q qqq q qq qq qq qq qq q qqq qqqqq qqqqqqq qqq q q qqq q q qqqqqq qq q qq qq q qq q q q qq q q qq qqq q q q q q qq qq q qq qq q q q q qqq q q q qq q q q q q q q q q q qq q q q q q q qq q qq q q q qq qq q qq q q q qqqqqq q qq q qqqqqqq q q q q q qqq qqqqqq qqq q q qqq qq q qq qq q qqqq q q q q q qq q q q qq qqq qq q q q q q q qqq q qqqq q qq qq q q qqq qq q q q qq q qq q qq q q q q q q q q q qq q q q q q q qq q q q q q q q qq q qq q qqq q q q q q qqq q q q q q qqq q qqq qq q q qq q qq q q qqq qq qq q qq q q qq qq q q qqq qqqqqqqqqq q qq qq qqq qqqq qqqq qq q q qqqqqqqq q q qq q qqq q q q qq q q q qq q qq qqq q q q qq q q q q q q q qq q q q q qq q q q q q q qq q q qq q qqqqq qqqq qqq qqq qqq qqq qqqq qq qq qqq q q qq q qqq q q q qq qq q qqqq qq q q q qq q q qqq q qqqq q q q q q qqq qq qq q qq q q qq qq q q q q q qq q q qq q q q q qqqq q q qq qq q q q q q q q q q q qqq q qq q qq q qq q q q q q qq q qq qq qq q q qq q qq q qqq qq qq q qqq qq qq q qq qqq q qq q qqq qqqqq q qqq q q q q q qqq q qqq q qq q qq q q q q q qq q qq π = π+1 ˆ ˆ q qq q q qq q q qq q qq q qqq q q qq q qqq qq q q q q q q q qq qq q qqqq qq qqq q qq q q qq q q q qqqqqq q q q q q qqqqqq qq q qq q q q q q qqq qqqq q qqq q q q q qq qqqq qq qq q qq q q q q q qq q q q qq qqq qq qq q q q qq qqq q q q qq q qqq q q q qq qq q qq q q q q q qq qq q q q q qq q qqq q qq qq q q qq q qq q q q q q q qq qq q q q q q q q qq q qq qqqqq q q qq qqqq qq q q qqq q qq q q q q q q q qqq qqqq qqq qq qq q qqq qq q q q q q qqq qqqqqqqq q qqqq qq qqqqq qqqq qqqqq qq qqqq qq qqqq qq qqq q q qqqqqq qq q q qq q q qq qq qqq q qq q qq q q q q qq qq q q q q qq q qq q q q qq q qqqq q q qqqqqqq qqq q qqqq qqqq q qqq q qq qqq q qqq q qq qq q qqq q q qq q q q qq q qq qq q q qq q q q qq q q qq qqq q qq q q qq q q q qq q q qq qq q qqq q q q q q q q qq q q q qq qq q qq qq q q qq q q qq qq q q q q q q q q q q q qq qq q q q q qq q q q q q q q q qq q q qqq q q q q q qqqq q q q q q qq qq q qq q q q qq q qq qq q q qqq q q q q q qqq q q q q qqq q q q qq qq q q q q q qq qq q q q q qqqqqqqqqqq q qqqq q qqq qqqqq qqqqq qqqqq qqq qq q q qq q q qqq qqqq qqq qqq q q q q q q q qq qq qq qq qq qq qqq q q q q qqqq q qq qqqqqq qqqq q q q q q qqq q q q qq q q qq q q qqqq q q q q q q q q q qq qqq qq q q q qqq qq q q qq q q q qq qq q qq q q qqq qq qq q q q q q q qqq qq qqq q q q qq q q q q q qq qq qq q q q qqq q q q q q q q q q q q q q q q qq qqqqq q q qqq q qqq q q qqqq q qqq q qq q q q q qqq qqqqq q q q q qq q q q qq q q qq q q q q qq q q qq q qq q q q q q q q q qq qq q q q q q q qq qqq q qqqq q q qqqqq q qqq qq q qq q q qqqqq qqqq q qq qqqqqqqqqqq qq q qq qq qq q qq q q q qq q q qq q q q q end if q q q qq q qq q qqq q q q q q qq qq qq q q qqq q q q qq q qq q q q q qq q q q q qq q qqq q q qq q q q qq qq q q q q qq q q q qq q q qq q q q q qq q q qqq q qqq q q q qq q q qqq q q qqqqq qq q qq qqqq qqqq q qq q qqq q q q q q q q q q q q q qqq qq qq qqq q q q qq q q q q q q q qq qq qq qq qqqqq q qqq q q qq q q qqqq q q qq qqqqqq q qq q q qqq qqq q q q q qq q q q qqqq q q q q q q q qq q q q q q q qq q q qq q q q q q qq q q qq q q q q qqqq q qq q qqq qq q qq qqq q qq q q qqq q q q q q q q q q q q qq q q q qq q q q q q q q q qq q qq qq q q q q qq qq q q q q q q q q q q q qq q q q qq q qqq q q qqq q q q q qq q q q qq q q q q qq q qq q qq q q qq q q q q qqqqqq q qqqqq q qq q qqqqq qqq qqq q q q q q qqqq q qqqqqq qq qqq qqqqq qq qq q qq q q qq q q qq q q q q q q q qq qq q q q qq qq qqq q q q qq qq qqq qqq qqq qqqq q q qq qq q q qqqqq qq qq qq q q qqq qqqqqq qqq qq q q qq q qq q q q q q qq q q qqqq qq qqqq q q q q qqqq qqqq qqqqq qqq qqqqqqqqqq qqqqqq q qqq qq q qqqq q q qqqqq qq q q q qq q q q q qq qqq q q q q q q q q qq qqq q q q qq q q q qqq qq q qqqqq qqq q qqqq qq q qq qq qqqqq qq q qq qqq q q q q q qq q qq q q qqq qq qq qq q qqq q q q q q q q qqq qq q qq q q q q q q q q q q q q q q qqq q q q q q q qq qq q q q q q q q q q qq qq q q qqq q q q q end for return 4*ˆ /N π 10,000 simulations
- 28. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Approximating π My daughter’s pseudo-code: N=1000 π=0 ˆ for I=1,N do X=RDN(1), Y=RDN(1) if X2 + Y2 < 1 then π = π+1 ˆ ˆ end if end for return 4*ˆ /N π 106 simulations
- 29. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Distributions that differ from uniform distributions Proble Given a probability distribution with density f , how can we produce randomness according to f ?! f (x) implemented algorithms in a resident software only available for common distributions new distributions may require fast resolution x no approximation allowed an arbitrary density
- 30. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Distributions that differ from uniform distributions Proble Given a probability distribution with density f , how can we produce randomness according to f ?! f (x) implemented algorithms in a resident software only available for common distributions new distributions may require fast resolution x no approximation allowed an arbitrary density
- 31. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means The Accept-Reject Algorithm (remember π?!) Given a probability distribution with density f , how can we produce randomness according to f ?! The uniform distribution on the sub-graph region Sf = {(x, u); 0 ≤ u ≤ f (x)} produces a marginal distribution f (x) in x with density f . (”Fundamental theorem of simulation”) x
- 32. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means The Accept-Reject Algorithm (remember π?!) Given a probability distribution with density f , how can we produce randomness according to f ?! q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q In practice, this means we are q q q q q q q q q q q q q q back to throwing dots on a box q q q q q q q q q q q q q q q q q containing q q q q q f (x) q q q q q q q q q q q q q q q q Sf = {(x, u); 0 ≤ u ≤ f (x)} q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q and counting those inside! q q q q q q q q q q q q q q q q q q q q q q q q q q q q q x
- 33. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means The Accept-Reject Algorithm Refinement of the above through construction of a proper box: 1. Find a hat on f , i.e. a density g that can be simulated and such that sup f (x) g (x) = M < ∞ x 2. Generate dots on the subgraph of g , i.e. Y1 , Y2 , . . . ∼ g , and U1 , U2 , . . . ∼ U ([0, 1]) 3. Accept only the Yk ’s such that Uk ≤ f (Yk )/Mg (Yk )
- 34. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means The Accept-Reject Algorithm Refinement of the above through construction of a proper box: 1. Find a hat on f , i.e. a density g that can be simulated and such that sup f (x) g (x) = M < ∞ x 2. Generate dots on the subgraph of g , i.e. Y1 , Y2 , . . . ∼ g , and U1 , U2 , . . . ∼ U ([0, 1]) 3. Accept only the Yk ’s such that Uk ≤ f (Yk )/Mg (Yk )
- 35. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means The Accept-Reject Algorithm Refinement of the above through construction of a proper box: 1. Find a hat on f , i.e. a density g that can be simulated and such that sup f (x) g (x) = M < ∞ x 2. Generate dots on the subgraph of g , i.e. Y1 , Y2 , . . . ∼ g , and U1 , U2 , . . . ∼ U ([0, 1]) 3. Accept only the Yk ’s such that Uk ≤ f (Yk )/Mg (Yk )
- 36. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means The Accept-Reject Algorithm Refinement of the above through construction of a proper box: q qq q q qq q q q qq q q q q q qq q q q q q q q qq q q q q q q qq q q q q qq qq qq q q q q q q q qq q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q qqq q q q q q q q q q q q q q q q qq q q q qq q q q q qq q q q q qq q q q q q q q q qq q q q q qqq qq q q q q q q q q q qq q q q qq q q q q q q qqq q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q qq qq q q qq q q q q q q q q qq q q qq qq q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q qq q q q q q q q qq q qq q q qq q qq q q qq q q q qq qq q q q q q q q q q q q f (x) q q q q q q q q q q qqq q q q qq q qq q q q q q q qq q q q q q q q q qq q q qq q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q qq q q q qq q q q q q q qq q q q q qq q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq qq q qq qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q q qq q q q q q q q q q q q q q qq q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq qq q q q q q q q qq q q q q qq q qq q q q q q q q qqq q q qq q q q q qq q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q qq q qq q q q q q qqq q q q q q q qq q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q qq q q q q q qq q q qq q q q q q q q q qq q q q q q q q q qq q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q qq q q qq q q q q q q qq q q q q qq q qq q q q q q q q q q q q q q q q q q qq q qqq qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q x
- 37. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Properties Does not require normalisation constant of f Does not require an exact upper bound M Requires on average M Yk ’s for one simulated X (efficiency measure)
- 38. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Properties Does not require normalisation constant of f Does not require an exact upper bound M Requires on average M Yk ’s for one simulated X (efficiency measure)
- 39. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Properties Does not require normalisation constant of f Does not require an exact upper bound M Requires on average M Yk ’s for one simulated X (efficiency measure)
- 40. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Slice sampler There exists other ways of exploitating the fundamental idea of simulation over the subgraph:
- 41. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Slice sampler There exists other ways of exploitating the fundamental idea of simulation over the subgraph: If direct uniform simulation on Sf = {(u, x); 0 ≤ u ≤ f (x)} is too complex [because of unavailable hat] use instead a random walk on Sf :
- 42. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Slice sampler There exists other ways of exploitating the fundamental idea of simulation over the subgraph: this means doing random jumps in vertical then horizontal direction, accounting for the boundaries 0 ≤ u ≤ f (x), i.e. U(0, 1) f (x) ≥ u, i.e. x ∼ US(u)
- 43. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Slice sampler Slice sampler algorithm For t = 1, . . . , T when at (x (t) , ω (t) ) simulate 1. ω (t+1) ∼ U[0,f (x (t) )] 2. x (t+1) ∼ US (t+1) , where S (t+1) = {y ; f (y ) ≥ ω (t+1) }.
- 44. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Slice sampler Slice sampler algorithm For t = 1, . . . , T 0.010 when at (x (t) , ω (t) ) simulate 0.008 1. ω (t+1) ∼ U[0,f (x (t) )] 0.006 y 2. x (t+1) ∼ US (t+1) , where S (t+1) = {y ; f (y ) ≥ ω (t+1) }. 0.004 0.002 0.000 0.0 0.2 0.4 0.6 0.8 1.0 x
- 45. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Slice sampler Slice sampler algorithm For t = 1, . . . , T 0.010 when at (x (t) , ω (t) ) simulate 0.008 1. ω (t+1) ∼ U[0,f (x (t) )] 0.006 y 2. x (t+1) ∼ US (t+1) , where S (t+1) = {y ; f (y ) ≥ ω (t+1) }. 0.004 0.002 0.000 0.0 0.2 0.4 0.6 0.8 1.0 x
- 46. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Slice sampler Slice sampler algorithm For t = 1, . . . , T 0.010 when at (x (t) , ω (t) ) simulate 0.008 1. ω (t+1) ∼ U[0,f (x (t) )] 0.006 2. x (t+1) ∼ US (t+1) , where y 0.004 S (t+1) = {y ; f (y ) ≥ ω (t+1) }. 0.002 0.000 0.0 0.2 0.4 0.6 0.8 1.0 x
- 47. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Slice sampler Slice sampler algorithm For t = 1, . . . , T 0.010 when at (x (t) , ω (t) ) simulate 0.008 1. ω (t+1) ∼ U[0,f (x (t) )] 0.006 2. x (t+1) ∼ US (t+1) , where y 0.004 S (t+1) = {y ; f (y ) ≥ ω (t+1) }. 0.002 0.000 0.0 0.2 0.4 0.6 0.8 1.0 x
- 48. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Slice sampler Slice sampler algorithm For t = 1, . . . , T 0.010 when at (x (t) , ω (t) ) simulate 0.008 1. ω (t+1) ∼ U[0,f (x (t) )] 0.006 2. x (t+1) ∼ US (t+1) , where y 0.004 S (t+1) = {y ; f (y ) ≥ ω (t+1) }. 0.002 0.000 0.0 0.2 0.4 0.6 0.8 1.0 x
- 49. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Slice sampler Slice sampler algorithm For t = 1, . . . , T 0.010 when at (x (t) , ω (t) ) simulate 0.008 1. ω (t+1) ∼ U[0,f (x (t) )] 0.006 2. x (t+1) ∼ US (t+1) , where y 0.004 S (t+1) = {y ; f (y ) ≥ ω (t+1) }. 0.002 0.000 0.0 0.2 0.4 0.6 0.8 1.0 x
- 50. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Slice sampler Slice sampler algorithm For t = 1, . . . , T 0.010 when at (x (t) , ω (t) ) simulate 0.008 1. ω (t+1) ∼ U[0,f (x (t) )] 0.006 2. x (t+1) ∼ US (t+1) , where y 0.004 S (t+1) = {y ; f (y ) ≥ ω (t+1) }. 0.002 0.000 0.0 0.2 0.4 0.6 0.8 1.0 x
- 51. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means The Metropolis–Hastings algorithm Further generalisation of the fundamental idea to situations when the slice sampler cannot be easily implemented
- 52. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means The Metropolis–Hastings algorithm Further generalisation of the fundamental idea to situations when the slice sampler cannot be easily implemented Idea Create a sequence (Xn ) such that, for n ‘large enough’, the density of Xn is close to f , only using a ‘local’ knowledge of f ... This is the domain of Markovian simulation methods: there is a Markov dependence between the Xn ’s Markov
- 53. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means The Metropolis–Hastings algorithm Further generalisation of the fundamental idea to situations when the slice sampler cannot be easily implemented ‘local’ exploration can produce ‘global’ vision: [ c Civilization 5]
- 54. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means The Metropolis–Hastings algorithm (2) If f is the density of interest, we pick a proposal conditional density q(y |xn ) such that it connects with the current value xn it is easy to simulate it is positive everywhere f is positive
- 55. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means The Metropolis–Hastings algorithm (2) If f is the density of interest, we pick a proposal conditional density q(y |xn ) such that it connects with the current value xn it is easy to simulate it is positive everywhere f is positive
- 56. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Random walk Metropolis–Hastings Proposal Yt = Xn + εn , where εn ∼ g , independent from Xn , and g symmetrical
- 57. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Random walk Metropolis–Hastings Proposal Yt = Xn + εn , where εn ∼ g , independent from Xn , and g symmetrical Motivation local perturbation of Xn / myopic exploration of its neighbourhood Yn accepted or rejected depending on the relative values of f (Xn ) and f (Yn )
- 58. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Random walk Metropolis–Hastings Proposal Yt = Xn + εn , where εn ∼ g , independent from Xn , and g symmetrical
- 59. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Random walk Metropolis–Hastings Resulting algorithm Random walk Metropolis–Hastings Starting from X (t) = x (t) 1. Generate Yt ∼ g (y − x (t) ) 2. Take f (Yt ) Y with proba. min 1, , (t+1) t X = f (x (t) ) (t) x otherwise
- 60. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Properties Always accepts higher point and sometimes lower points (similar to gradient algorithms) Depends on the dispersion de g Average probability of acceptance = min{f (x), f (y )}g (y − x) dxdy close to 1 if g has a small variance [Danger!] far from 1 if g has a large variance [Re-Danger!]
- 61. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Properties Always accepts higher point and sometimes lower points (similar to gradient algorithms) Depends on the dispersion de g Average probability of acceptance = min{f (x), f (y )}g (y − x) dxdy close to 1 if g has a small variance [Danger!] far from 1 if g has a large variance [Re-Danger!]
- 62. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Properties Always accepts higher point and sometimes lower points (similar to gradient algorithms) Depends on the dispersion de g Average probability of acceptance = min{f (x), f (y )}g (y − x) dxdy close to 1 if g has a small variance [Danger!] far from 1 if g has a large variance [Re-Danger!]
- 63. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Properties scale=1
- 64. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Properties scale=2
- 65. Simulation: an ubiquitous tool for statistical computation Producing randomness by deterministic means Properties scale=3
- 66. Simulation: an ubiquitous tool for statistical computation Monte Carlo principles General purpose of Monte Carlo methods Given a probability density f known up to a normalizing constant, ˜ f (x) ∝ f (x), and an integrable function h, compute ˜ h(x)f (x)dx I(h) = h(x)f (x)dx = ˜ f (x)dx when ˜ h(x)f (x)dx is intractable. [Remember π!!]
- 67. Simulation: an ubiquitous tool for statistical computation Monte Carlo principles General purpose of Monte Carlo methods Given a probability density f known up to a normalizing constant, ˜ f (x) ∝ f (x), and an integrable function h, compute ˜ h(x)f (x)dx I(h) = h(x)f (x)dx = ˜ f (x)dx when ˜ h(x)f (x)dx is intractable. [Remember π!!]
- 68. Simulation: an ubiquitous tool for statistical computation Monte Carlo principles Monte Carlo basics Generate a pseudo-random sample x1 , . . . , xN from f and estimate I(h) by N IN (h) = N −1 ˆ MC h(xi ). i=1 and let N grow to infinity...
- 69. Simulation: an ubiquitous tool for statistical computation Monte Carlo principles Monte Carlo basics Generate a pseudo-random sample x1 , . . . , xN from f and estimate I(h) by N IN (h) = N −1 ˆ MC h(xi ). i=1 and let N grow to infinity... A. Doucet, C. Andrieu, X, A. Philippe, J. Rosenthal, E. Moulines
- 70. Simulation: an ubiquitous tool for statistical computation Monte Carlo principles Monte Carlo basics Generate a pseudo-random sample x1 , . . . , xN from f and estimate I(h) by N IN (h) = N −1 ˆ MC h(xi ). i=1 and let N grow to infinity...
- 71. Simulation: an ubiquitous tool for statistical computation Monte Carlo principles Monte Carlo basics Generate a pseudo-random sample x1 , . . . , xN from f and estimate I(h) by N IN (h) = N −1 ˆ MC h(xi ). i=1 and let N grow to infinity... Caveat Often impossible or inefficient to simulate directly from f
- 72. Simulation: an ubiquitous tool for statistical computation Monte Carlo principles Importance Sampling For proposal distribution with density q(x), alternative representations I(h) = h(x){f /q}(x)q(x)dx Principle Generate an iid sample x1 , . . . , xN ∼ q and estimate I(h) by N Iq,N (h) = N −1 ˆ IS h(xi ){f /q}(xi ). i=1
- 73. Simulation: an ubiquitous tool for statistical computation Monte Carlo principles Importance Sampling For proposal distribution with density q(x), alternative representations I(h) = h(x){f /q}(x)q(x)dx Principle Generate an iid sample x1 , . . . , xN ∼ q and estimate I(h) by N Iq,N (h) = N −1 ˆ IS h(xi ){f /q}(xi ). i=1
- 74. Simulation: an ubiquitous tool for statistical computation Simulated annealing Optimisation problems A (genuine) puzzle During a dinner with 20 couples sitting at four tables with ten seats, everyone wants to share a table with everyone. The assembly decides to switch seats after each serving towards this goal. What is the minimal number of servings needed to ensure that every couple shared a table with every other couple at some point? And what is the optimal switching strategy? [http://xianblog.wordpress.com/2012/04/12/not-le-monde-puzzle/] solution to the puzzle
- 75. Simulation: an ubiquitous tool for statistical computation Simulated annealing Stochastic minimisation Example (Egg-box function) Consider the function h(x, y ) = (x sin(20y ) + y sin(20x))2 cosh(sin(10x)x) + (x cos(10y ) − y sin(10x))2 cosh(cos(20y )y ) , to be minimised.
- 76. Simulation: an ubiquitous tool for statistical computation Simulated annealing Stochastic minimisation Example (Egg-box function) Consider the function h(x, y ) = (x sin(20y ) + y sin(20x))2 cosh(sin(10x)x) + (x cos(10y ) − y sin(10x))2 cosh(cos(20y )y ) , to be minimised. (I know that the global minimum is 0 for (x, y ) = (0, 0).)
- 77. Simulation: an ubiquitous tool for statistical computation Simulated annealing Stochastic minimisation Example (Egg-box function) Consider the function h(x, y ) = (x sin(20y ) + y sin(20x))2 cosh(sin(10x)x) + (x cos(10y ) − y sin(10x))2 cosh(cos(20y )y ) , to be minimised. (I know that the global minimum is 0 for (x, y ) = (0, 0).)
- 78. Simulation: an ubiquitous tool for statistical computation Simulated annealing Stochastic minimisation Example (egg-box function (2)) Instead of solving the first order equations ∂h(x, y ) ∂h(x, y ) = 0, =0 ∂x ∂y and of checking that the second order conditions are met, we can generate a random sequence in R2 αj θj+1 = θj + ∆h(θj , βj ζj ) ζj 2βj
- 79. Simulation: an ubiquitous tool for statistical computation Simulated annealing Stochastic minimisation Example (egg-box function (2)) we can generate a random sequence in R2 αj θj+1 = θj + ∆h(θj , βj ζj ) ζj 2βj where the ζj ’s are uniform on the unit circle x 2 + y 2 = 1; ∆h(θ, ζ) = h(θ + ζ) − h(θ − ζ); (αj ) and (βj ) converge to 0
- 80. Simulation: an ubiquitous tool for statistical computation Simulated annealing Calibration parameters choice 1 2 3 4 αj 1/ log(j + 1) 1/100 log(j + 1) 1/(j + 1) 1/(j + 1) βj 1/ log(j + 1).1 1/ log(j + 1).1 1/(j + 1).5 1/(j + 1).1 α ↓ 0 slowly, j αj = ∞ β ↓ 0 more slowly, 2 j (αj /βj ) < ∞ Scenarios 1-2: not enough energy Scenarios 3-4: good [ c George Casella 1951–2012
- 81. Simulation: an ubiquitous tool for statistical computation Simulated annealing The traveling salesman problem A classical allocation problem: Salesman who needs to visit n cities in a row Traveling costs between pairs of cities are known Search of the optimal circuit Procter & Gamble competition, 1962
- 82. Simulation: an ubiquitous tool for statistical computation Simulated annealing The traveling salesman problem A classical allocation problem: Salesman who needs to visit n cities in a row Traveling costs between pairs of cities are known Search of the optimal circuit Procter & Gamble competition, 1962
- 83. Simulation: an ubiquitous tool for statistical computation Simulated annealing The traveling salesman problem A classical allocation problem: Salesman who needs to visit n cities in a row Traveling costs between pairs of cities are known Search of the optimal circuit Procter & Gamble competition, 1962
- 84. Simulation: an ubiquitous tool for statistical computation Simulated annealing An NP-complete problem The traveling salesman problem is an example of mathematical problems that require explosive resolution times
- 85. Simulation: an ubiquitous tool for statistical computation Simulated annealing An NP-complete problem The traveling salesman problem is an example of mathematical problems that require explosive resolution times Number of possible circuits n! and exact solutions available in O(2n ) time
- 86. Simulation: an ubiquitous tool for statistical computation Simulated annealing An NP-complete problem The traveling salesman problem is an example of mathematical problems that require explosive resolution times Numerous practical consequences (networks, integrated circuit design, genomic sequencing, &tc.)
- 87. Simulation: an ubiquitous tool for statistical computation Simulated annealing An NP-complete problem The traveling salesman problem is an example of mathematical problems that require explosive resolution times Number of possible circuits n! and exact solutions available in O(2n ) time Exact solution for 15, 112 German cities found in 2001 in 22.6 CPU years.
- 88. Simulation: an ubiquitous tool for statistical computation Simulated annealing An NP-complete problem The traveling salesman problem is an example of mathematical problems that require explosive resolution times Number of possible circuits n! and exact solutions available in O(2n ) time Exact solution for the 24, 978 Sweedish cities found in 2004 in 84.8 CPU years.
- 89. Simulation: an ubiquitous tool for statistical computation Simulated annealing Resolution via simulation The simulated annealing algorithm: name is borrowed from metallurgy metal manufactured by a slow decrease of temperature (annealing) stronger than when manufactured by fast decrease [ c Joachim Robert, ca. 2006]
- 90. Simulation: an ubiquitous tool for statistical computation Simulated annealing Resolution via simulation Repeat Random modifications of parts of the original circuit with cost C0 Evaluation of the cost C of the new circuit Acceptation of the new circuit with probability C0 − C exp ∧1 T [Metropolis et al., 1953]
- 91. Simulation: an ubiquitous tool for statistical computation Simulated annealing Resolution via simulation Repeat Random modifications of parts of the original circuit with cost C0 Evaluation of the cost C of the new circuit Acceptation of the new circuit with probability C0 − C exp ∧1 T T , temperature, is progressively reduced [Metropolis et al., 1953]
- 92. Simulation: an ubiquitous tool for statistical computation Simulated annealing A family meeting (1) Recall the table puzzle Definition of a target function Set the story of the 20 couple tables during the 6 courses as a (20, 6) matrix, e.g., 1 1 3 2 4 2 2 1 3 3 2 4 S = . . . . . . . . . . . . . . . . . . 3 2 2 2 4 4 and define a penalty function as the number of missed encounters
- 93. Simulation: an ubiquitous tool for statistical computation Simulated annealing A family meeting (1) Recall the table puzzle Definition of a target function T=sample(rep(1:4,5)) for (i in 2:6) T=cbind(T,sample(rep(1:4,5))) meet=function(T){ M=outer(T[,1],T[,1],"==") for (i in 2:6) M=M+outer(T[,i],T[,i],"==") M } penalty=function(M){ sum(M==0) } penat=penalty(meet(T))
- 94. Simulation: an ubiquitous tool for statistical computation Simulated annealing A family meeting (2) Random switch of couples Pick two couples [among the 20 couples] at random with probabilities proportional to the number of other couples they have not seen prob=apply(meet(T)==0,1,sum) switch their respective position during one of the 6 courses accept the switch with Metropolis–Hsatings probability log(runif(1))<(penalty(old)-penalty(new))/gamma
- 95. Simulation: an ubiquitous tool for statistical computation Simulated annealing A family meeting (2) Random switch of couples For instance, propose toreplace 1 1 3 2 4 2 1 1 3 2 2 2 2 1 3 3 2 4 2 1 3 3 4 4 S = . . . . . . with S = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 2 2 4 4 3 2 2 2 4 4
- 96. Simulation: an ubiquitous tool for statistical computation Simulated annealing A family meeting (2) Random switch of couples for (t in 1:N){ prop=sample(1:20,2,prob=apply(meet(T)==0,1,sum)) cour=sample(1:6,1) Tp=T Tp[prop[1],cour]=T[prop[2],cour] Tp[prop[2],cour]=T[prop[1],cour] penatp=penalty(meet(Tp)) if (log(runif(1))<(penat-penatp)/gamma){ T=Tp penat=penatp} }
- 97. Simulation: an ubiquitous tool for statistical computation Simulated annealing A family meeting (3) Solution T [1,] 1 4 3 2 2 3 [2,] 1 2 4 3 4 4 [3,] 3 2 1 4 1 3 [4,] 1 2 3 1 1 1 [5,] 4 2 4 2 3 3 [6,] 2 4 1 2 4 1 [7,] 4 3 1 1 2 4 [8,] 1 3 2 4 3 1 [9,] 3 3 3 3 4 3 [10,] 4 4 2 3 1 1 [11,] 1 1 1 3 3 2 [12,] 3 4 4 1 3 2 [13,] 4 1 3 4 4 2 [14,] 2 4 3 4 3 4 [15,] 2 3 4 2 1 2 [16,] 2 2 2 3 2 2 [17,] 2 1 2 1 4 3 [18,] 4 3 1 1 2 4 [19,] 3 1 4 4 2 1 [20,] 3 1 2 2 1 4 [ c http://www.metrolyrics.com]
- 98. Simulation: an ubiquitous tool for statistical computation Simulated annealing Solving sudokus Solving a Sudoku grid as a minimisation problem: [ c Dan Rice Sudoku blog]
- 99. Simulation: an ubiquitous tool for statistical computation Simulated annealing Solving sudokus Given a partly filled Sudoku grid (with a single solution), define a random Sudoku grid by filling the empty slots at random s=matrix(0,ncol=9,nrow=9) s[1,c(1,6,7)]=c(8,1,2) s[2,c(2:3)]=c(7,5) s[3,c(5,8,9)]=c(5,6,4) s[4,c(3,9)]=c(7,6) s[5,c(1,4)]=c(9,7) s[6,c(1,2,6,8,9)]=c(5,2,9,4,7) s[7,c(1:3)]=c(2,3,1) s[8,c(3,5,7,9)]=c(6,2,1,9)
- 100. Simulation: an ubiquitous tool for statistical computation Simulated annealing Solving sudokus Given a partly filled Sudoku grid (with a single solution), define a random Sudoku grid by filling the empty slots at random define a penalty function corresponding to the number of missed constraints #local score scor=function(i,s){ a=((i-1)%%9)+1 b=trunc((i-1)/9) boxa=3*trunc((a-1)/3)+1 boxb=3*trunc(b/3)+1 return(sum(s[i]==s[9*b+(1:9)])+ sum(s[i]==s[a+9*(0:8)])+ sum(s[i]==s[boxa:(boxa+2),boxb:(boxb+2)])-3) }
- 101. Simulation: an ubiquitous tool for statistical computation Simulated annealing Solving sudokus Given a partly filled Sudoku grid (with a single solution), define a random Sudoku grid by filling the empty slots at random define a penalty function corresponding to the number of missed constraints fill “deterministic” slots make simulated annealing moves # random moves on the sites i=sample((1:81)[as.vector(s)==0],sample(1:sum(s==0),1,pro=1/(1:sum(s==0)))) for (r in 1:length(i)) prop[i[r]]=sample((1:9)[pool[i[r]+81*(0:8)]],1) if (log(runif(1))/lcur<tarcur-target(prop)){ nchange=nchange+(tarcur>target(prop)) cur=prop points(t,tarcur,col="forestgreen",cex=.3,pch=19) tarcur=target(cur) }
- 102. Simulation: an ubiquitous tool for statistical computation Simulated annealing Solving sudokus many possible variants rather slow may get stuck on a penalty of 2 does not compete with nonrandom solvers